## Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments

Venue: | Ann. of Math |

Citations: | 38 - 8 self |

### BibTeX

@ARTICLE{Dimca_hypersurfacecomplements,,

author = {Ru Dimca and Stefan Papadima},

title = {Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments},

journal = {Ann. of Math},

year = {},

pages = {473--507}

}

### OpenURL

### Abstract

The interplay between geometry and topology on complex algebraic varieties is a classical theme that goes back to Lefschetz [L] and Zariski [Z] and is always present on the scene, see for instance the work by Libgober [Li]. In this paper we study complements of hypersurfaces, with a special attention to the case of hyperplane arrangements as discussed

### Citations

223 |
Rational homotopy theory
- Quillen
- 1969
(Show Context)
Citation Context ...le arrangements A for which π2M(A) is an infinitely generated free abelian group; see [PS, Remark 6.9 and Theorem 6.11]. It follows that ˜ M(A) is rationally an infinite wedge of 2-spheres. Following =-=[Q]-=-, the rational homotopy Lie algebra (under Whitehead product) π>1M(A) ⊗ Q is a free graded Lie algebra on infinitely many generators [Hil]. 6. Some higher homotopy groups of arrangements In this secti... |

206 |
H.: Arrangements of hyperplanes
- Orlik, Terao
- 1992
(Show Context)
Citation Context ...responding principal open set M = D(Q) = P n \ ∪i=1,dHi. The topology of the hyperplane arrangement complement M is a central object of study in the theory of hyperplane arrangements, see Orlik-Terao =-=[OT1]-=-. As a consequence of Theorem 1. we prove the following. 2sCorollary 4. For any projective arrangement A as above one has In particular, the following are equivalent. (i) the morphism grad(Q) is domin... |

195 |
Stratified Morse Theory
- Goresky, MacPherson
- 1988
(Show Context)
Citation Context ...h Theorem 1. and Theorem 3. follow from the results by Hamm in [H]. In the case of Theorem 1. the homotopy type claim is a direct consequence from [H], Theorem 5. and also from Goresky and MacPherson =-=[GM]-=-, Theorem 4.1, the new part being the relation between the number of n-cells and the degree of the gradient map grad(h). We establish this equality by using polar curves, see section 2. On the other h... |

128 |
Combinatorics and topology of complements of hyperplanes
- Orlik, Solomon
- 1980
(Show Context)
Citation Context ...⊂ A, which is a q-circuit (i.e. minimally dependent), one associates the element eB ∈ ∧q (e1, . . ., en), eB := ei1 · · ·eiq . Define next (8) ∂eB := q∑ (−1) r−1 ei1 · · ·êir · · ·eiq . r=1 One knows =-=[OS]-=- that there is a natural algebra isomorphism between A ∗ (A) := ∧ ∗ (e1, . . ., en) / ideal (∂eB | B ⊂ A a circuit) and the cohomology algebra H ∗ (M ′ , Z). We are thus naturally led to define c(A) b... |

69 |
The lower central series of a fiber-type arrangement
- Falk, Randell
- 1985
(Show Context)
Citation Context ... speaking, that A and Â have the same intersection lattice, up to rank k + 1; see §5(1) for the precise definition.) The general position arrangements from [Hat] and the fiber-type aspheric ones from =-=[FR]-=- belong to the hypersolvable class from [JP]. Consequently ([JP1]), they all are 2generic sections of fiber-type arrangements. Note also that the fundamental groups of hypersolvable complements exhibi... |

61 |
On the problem of existence of algebraic functions of two variables possessing a given branch curve
- Zariski
- 1929
(Show Context)
Citation Context ...em; this is an object that records information on the type of singularities, and on their position (on the arrangement). Our examples above may thus be viewed as higher analogues of Zariski’s results =-=[Z]-=- on the non-influence of the position in P 2 of some singularities (e.g. nodes), on π1 of the complement of a plane curve. On the other hand, our examples should be compared to those from [Ry], which ... |

48 |
The topology of complex projective varieties after
- Lamotke
- 1981
(Show Context)
Citation Context ...e integer (Ft, ΓH) is called the polar invariant of the hypersurface Ft and is denoted by P(Ft). Note that P(Ft) corresponds exactly to the classical notion of class of a projective hypersurface, see =-=[L]-=-. We think of a projective hyperplane H as above as the direction of an affine hyperplane H ′ = {x ∈ C n+1 |ℓ(x) = s} for s ∈ C. All the hyperplanes with the same direction form a pencil, and it is pr... |

48 | Obstructions to homotopy equivalences - Halperin, Stasheff - 1979 |

45 |
Sur les groupes de tresses
- Brieskorn
- 1973
(Show Context)
Citation Context ... see Remark 12 (ii). In the second part of our paper, we investigate the higher homotopy groups of complements of complex hyperplane arrangements (as π1-modules). By the classical work of 1Brieskorn =-=[B]-=- and Deligne [De], it is known that such a complement is often aspherical. The first explicit computation of nontrivial homotopy groups of this type has been performed by Hattori [Hat], in 1975. This ... |

42 | Singularities at Infinity and their Vanishing Cycles
- Siersma, Tibăr
- 1995
(Show Context)
Citation Context ...re on this equivalence see the beginning of the proof of Theorem 3.) is a topic under intense investigations, see for instance Cassou-Noguès and Dimca [CD], Hamm [H], Némethi [N12], Siersma and Tibăr =-=[ST]-=-, [T]. For all the proofs in this paper, the classical (local) theory is sufficient: indeed, all the objects being homogeneous, one can localize at the origin of 4sC n+1 in the standard way, see [D1].... |

39 |
Homotopy groups of the complements to singular hypersurfaces
- Libgober
- 1985
(Show Context)
Citation Context ... real affine subspaces; see [GM, p. 236]. The investigation of higher homotopy groups of complements of complex hypersurfaces (as π1-modules) is a very difficult problem. In the irreducible case, see =-=[Li]-=- for various results on the first nontrivial higher homotopy group. In the simplest non-irreducible case (hyperplane arrangements), the first explicit computation of nontrivial higher homotopy groups ... |

35 |
Intersection form for quasihomogeneous singularities
- Steenbrink
- 1977
(Show Context)
Citation Context ...ed from the lattice associated to the arrangement (see Corollary (2.3) and Remark (2.7) in [DL]) using the fact that the weight equivariant Euler polynomial of the µe-variety F is known, see [MO] and =-=[St]-=-. Example 13. In this example we explain why special care is needed when doing Morse theory on non compact manifolds as in [OT2] and [R]. Let’s start with a very simple case, where computations are ea... |

34 |
Isolated singularities defined by weighted homogeneous polynomials. Topology 9
- Milnor, Orlik
- 1970
(Show Context)
Citation Context ... F ∩ N is (n − 2)-connected (use the exact homotopy sequence of the pair (F, F ∩ N) and the fact that F is (n − 1)-connected since it is homotopy equivalent to a bouquet of (e − 1) n+1 spheres S n by =-=[MO]-=-) and has the homotopy type of a CW-complex of dimension ≤ (n − 1) (being an affine variety of dimension n − 1 see [H], [GM]). 4. Complements of hyperplane arrangements 12Proof of Lemma 5. We are goi... |

32 | On the fundamental group of the complement of a complex hyperplane arrangement, Funct
- Rybnikov
(Show Context)
Citation Context ... results [Z] on the non-influence of the position in P 2 of some singularities (e.g. nodes), on π1 of the complement of a plane curve. On the other hand, our examples should be compared to those from =-=[Ry]-=-, which show that, in general, the combinatorics does not determine the topology of an arrangement complement (not even the fundamental group) in analogy to the famous sextics with 6 cusps considered ... |

26 | Homology of iterated semi-direct products of free groups
- Cohen, Suciu
- 1994
(Show Context)
Citation Context ... A, by realizing it as a 2-generic section of a fiber-type arrangement Â. From the computational point of view, note also that the boundary maps in (2) may be explicitly computed by Fox calculus (see =-=[CS]-=-), while p and Π∗ from Theorem 15. are combinatorially determined, from the intersection lattice L(A); see [PS, Theorem 5.4 and Corollary 5.5]. Our next goal in this section is to analyse a natural si... |

26 |
On the homotopy groups of the union of spheres
- Hilton
- 1955
(Show Context)
Citation Context ... ˜ M(A) is rationally an infinite wedge of 2-spheres. Following [Q], the rational homotopy Lie algebra (under Whitehead product) π>1M(A) ⊗ Q is a free graded Lie algebra on infinitely many generators =-=[Hil]-=-. 6. Some higher homotopy groups of arrangements In this section, we are going to illustrate Theorem 15. on two simple classes of examples: iterated generic hyperplane sections of aspheric arrangement... |

24 |
Variétés polaires Locales et classes de Chern des variétés singulières
- LÊ, TEISSIER
- 1981
(Show Context)
Citation Context ...R]. 2. Polar curves, affine Lefschetz theory and degree of gradient maps The use of the local polar varieties in the study of singular spaces is already a classical subject, see Lê [Lê], Lê -Teissier =-=[LT]-=- and the references therein. Global polar curves in the study of the topology of polynomials (or, equivalently, the affine Lefschetz theory, for more on this equivalence see the beginning of the proof... |

24 |
Topology of C n minus a finite number of affine hyperplanes in general position
- Hattori
- 1975
(Show Context)
Citation Context ... higher homotopy group. In the simplest non-irreducible case (hyperplane arrangements), the first explicit computation of nontrivial higher homotopy groups has been performed by Hattori, 25 years ago =-=[Hat]-=-. This remained the only example of this kind in arrangement theory, until [PS] came out. We devote the last two sections (§§5-6) to explicit descriptions of (nontrivial) higher homotopy groups of arr... |

20 | Higher homotopy groups of complements of complex hyperplane arrangements
- Papadima, Suciu
(Show Context)
Citation Context ...f a CW-complex K whose number of k-cells equals bk(K) for all k ∈ N. The importance of this notion for the topology of hyperplane arrangements was recently discovered by S. Papadima and A. Suciu, see =-=[PS]-=- for various applications. The following result was independantly obtained by Randell, see [R], using similar techniques. Corollary 6. The complement M is a minimal space. It is easy to see that for n... |

17 |
Dimca: Singularities and Topology of Hypersurfaces, Universitext
- unknown authors
- 1992
(Show Context)
Citation Context ...c [Sz], where degrees of real polynomials play a similar role. Let f ∈ C[x0, ..., xn] be a homogeneous polynomial of degree e > 0 with global Milnor fiber F = {x ∈ C n+1 |f(x)) = 1}, see for instance =-=[D1]-=- for more on such varieties. Let g : F \ N → R be the function g(x) = h(x)h(x), where N = {x ∈ C n+1 |h(x)) = 0}. Then we have the following. Theorem 3. For any reduced homogeneous polynomial h ∈ C[x0... |

17 |
A generalization of fiber-type arrangements and a new deformation method, Topology 37
- Jambu, Papadima
- 1998
(Show Context)
Citation Context ...ection lattice, up to rank k + 1; see §5(1) for the precise definition.) The general position arrangements from [Hat] and the fiber-type aspheric ones from [FR] belong to the hypersolvable class from =-=[JP]-=-. Consequently ([JP1]), they all are 2generic sections of fiber-type arrangements. Note also that the fundamental groups of hypersolvable complements exhibit a rich structure: they are iterated almost... |

16 |
Calcul du nombre de cycles évanouissants d’une hypersurface complexe
- Lê
- 1973
(Show Context)
Citation Context ...andell’s preprint [R]. 2. Polar curves, affine Lefschetz theory and degree of gradient maps The use of the local polar varieties in the study of singular spaces is already a classical subject, see Lê =-=[Lê]-=-, Lê -Teissier [LT] and the references therein. Global polar curves in the study of the topology of polynomials (or, equivalently, the affine Lefschetz theory, for more on this equivalence see the beg... |

14 | SPANIER: Algebraic Topology - H - 1966 |

13 |
Lefschetz theorems for singular varieties
- Hamm
- 1983
(Show Context)
Citation Context ...btained from F ∩N by attaching |C(g)| cells of dimension n, where C(g) is the critical set of the Morse function g. We point out that both Theorem 1. and Theorem 3. follow from the results by Hamm in =-=[H]-=-. In the case of Theorem 1. the homotopy type claim is a direct consequence from [H], Theorem 5. and also from Goresky and MacPherson [GM], Theorem 4.1, the new part being the relation between the num... |

13 |
Stratification and flatness
- Hironaka
(Show Context)
Citation Context ... for h and for any other stratum, say S1 ⊂ h−1 (0), and any sequence of points qm ∈ S0 converging to q ∈ S1 such that the sequence of tangent spaces Tqm (h) has a limit T, then TqS1 ⊂ T, see Hironaka =-=[Hi]-=-, Corollary 1, page 248 (and note that the requirement of f proper in that Corollary is not necessary in our case, as any algebraic map can be compactified). Here and in the sequel, for a map φ : X → ... |

13 | Morse theory, Milnor fibers and minimality of hyperplane arrangements
- Randell
(Show Context)
Citation Context ...[CS]). It has been verified for arbitrary arrangements, up to k = 2, in [PS, Theorem 4.3]. Our next result establishes this property, in full generality. It was independantly obtained by Randell, see =-=[Ra]-=-, using similar techniques. (See, however, Example 13.) Corollary 6. Both complements, M(A) ⊂ P n and its cone, M ′ (A) ⊂ C n+1 , are minimal spaces. 5It is easy to see that for n > 1, the open set D... |

12 |
Asymptotic equisingularity and topology of complex hypersurfaces Int
- Tibăr
- 1998
(Show Context)
Citation Context ...this equivalence see the beginning of the proof of Theorem 3.) is a topic under intense investigations, see for instance Cassou-Noguès and Dimca [CD], Hamm [H], Némethi [N12], Siersma and Tibăr [ST], =-=[T]-=-. For all the proofs in this paper, the classical (local) theory is sufficient: indeed, all the objects being homogeneous, one can localize at the origin of 4sC n+1 in the standard way, see [D1]. Howe... |

11 | On the connectivity of complex affine hypersurfaces
- Dimca, Păunescu
(Show Context)
Citation Context ...3. Non-proper Morse Theory For the convenience of the reader we recall, in the special case we need, a basic result of Hamm, see [H], Proposition 3, with our addition concerning the condition (c0) in =-=[DP]-=-, see Lemma 3. and Example 2. The final claim on the number of cells to be attached is also standard, see for instance [ST] and [T]. Proposition 11. Let A be a smooth algebraic subvariety in C p with ... |

11 |
Stratification and flatness Real and Complex singularities, Nordic Summer School
- Hironaka
- 1976
(Show Context)
Citation Context ... for h and for any other stratum, say S1 ⊂ h−1 (0), and any sequence of points qm ∈ S0 converging to q ∈ S1 such that the sequence of tangent spaces Tqm (h) has a limit T, then TqS1 ⊂ T, see Hironaka =-=[Hi]-=-, Corollary 1, page 248 (and note that the requirement of f proper in that Corollary is not necessary in our case, as any algebraic map can be compactified). Here and in the sequel, for a map φ : X → ... |

9 |
Homotopy and group cohomology of arrangements
- Randell
- 1997
(Show Context)
Citation Context ... braid groups, indicate a highly nontrivial twisting of this product structure. At the same time, the iterated generic hyperplane sections, A, of essential aspheric arrangements, rank(A) − 1. Â, from =-=[R1]-=-, are also particular cases of k-generic sections, with k = For such a k-generic section A, Theorem 15. firstly says that the complement M(A) (M ′ (A)) is aspheric if and only if p = ∞, where p is a n... |

6 | Théorie de Lefschetz pour les variétés algébriques affines - Némethi |

5 | Lefschetz theory for complex affine varieties - Némethi |

3 |
Dimca: Topology of complex polynomials via polar curves
- Cassou-Noguès
- 1999
(Show Context)
Citation Context ...equivalently, the affine Lefschetz theory, for more on this equivalence see the beginning of the proof of Theorem 3.) is a topic under intense investigations, see for instance Cassou-Noguès and Dimca =-=[CD]-=-, Hamm [H], Némethi [N12], Siersma and Tibăr [ST], [T]. For all the proofs in this paper, the classical (local) theory is sufficient: indeed, all the objects being homogeneous, one can localize at the... |

3 |
Arrangements, Milnor fibers and polar curves
- Dimca
(Show Context)
Citation Context ...rom F ∩N by attaching |C(g)| cells of dimension n, where C(g) is the critical set of the Morse function g. In particular bn(F, F ∩ N) = |C(g)|. This paper represents a strengthening of the results in =-=[D2]-=- (in which the homological version of Theorem 1. and 3. above was proven). The author thanks Stefan Papadima for raising the question answered by Corollary 4 above and for lots of helpful comments. In... |

3 |
Lehrer: Purity and equivariant weight polynomials, dans le volume: Algebraic Groups and
- Dimca, I
- 1997
(Show Context)
Citation Context ...riction, see [OT1], p. 17, the obvious additivity of the Euler characteristics and, more subtly, the additivity of the top Betti numbers coming from the exact sequence (8) in [OT1], p. 20 or (3.8) in =-=[DL]-=-. 9sProof of Corollary 4. To complete this proof we only have to explain why the claims (ii) and (iii) are equivalent. If the projective arrangement is not essential, then using a projection onto P n−... |

3 |
On the Euler characteristic of complex algebraic varieties
- Szafraniec
- 1988
(Show Context)
Citation Context ...nstructible partitions, one obtains formulas for the Euler characteristic of any constructible set in terms of an alternating sum of degrees. This result should be compared with results by Szafraniec =-=[Sz]-=-, where degrees of real polynomials play a similar role. Let f ∈ C[x0, ..., xn] be a homogeneous polynomial of degree e > 0 with global Milnor fiber F = {x ∈ C n+1 |f(x)) = 1}, see for instance [D1] f... |

3 | Un théorème de Zariski du type de - Hamm, Trãng - 1973 |

2 | Morse theory, Milnor fibers and hyperplane arrangements, math.AT 0011101, version 2
- Randell
(Show Context)
Citation Context ...ion for the topology of hyperplane arrangements was recently discovered by S. Papadima and A. Suciu, see [PS] for various applications. The following result was independantly obtained by Randell, see =-=[R]-=-, using similar techniques. Corollary 6. The complement M is a minimal space. It is easy to see that for n > 1, the open set D(f) is not minimal for f generic of degree d > 1 (just use H1(D(f), Z) = Z... |

2 |
Stratified Morse Theory, Ergebnisse 14
- Goresky, MacPherson
- 1988
(Show Context)
Citation Context ... on CWstructures with geometrically controlled number of cells, on complements and Milnor fibers respectively. The case of hyperplane arrangements has received considerable attention, see for example =-=[GM]-=-, [OT1]. In this particular case, we obtain from Theorems 1. and 3., Corollaries 4. and 7. respectively. They may both be viewed as improving similar results, from [GM, Part III] and [OT2] respectivel... |

2 |
Papadima: Deformations of hypersolvable arrangements
- Jambu, S
(Show Context)
Citation Context ...o rank k + 1; see §5(1) for the precise definition.) The general position arrangements from [Hat] and the fiber-type aspheric ones from [FR] belong to the hypersolvable class from [JP]. Consequently (=-=[JP1]-=-), they all are 2generic sections of fiber-type arrangements. Note also that the fundamental groups of hypersolvable complements exhibit a rich structure: they are iterated almost-direct products of f... |

2 |
Dolgachev Polar Cremona transformations
- V
- 2000
(Show Context)
Citation Context ...ponding assertions at the level of cells. In addition, our results give a positive answer to Dolgachev’s conjecture on polar Cremona transformations as well as stronger versions of several results in =-=[Do]-=-, see Corollary 2., Corollary 4., and the discussion at the end of section 3. Corollary 6. (proved in §4) reveals a striking feature of complements of complex hyperplane arrangements, among hypersurfa... |

1 |
Stratifications and flatness,in
- Hironaka
- 1977
(Show Context)
Citation Context ...for h and for any other stratum, say S1 ⊂ h −1 (0), and any sequence of points qm ∈ S0 converging to q ∈ S1 such that the sequence of tangent spaces Tqm (h) has a limit T, then TqS1 ⊂ T, see Hironaka =-=[Hi]-=-, Corollary 1, page 248 (and note that the requirement of f proper in that Corollary is not necessary in our case, as any algebraic map can be compactified). Here and in the sequel, for a map φ : X → ... |

1 |
Arrangements, Milnor fibers and polar curves, math.AG/0011073, version 2
- Dimca
(Show Context)
Citation Context ...ticular bn(F, F ∩ N) = |C(g)| and the intersection F ∩ N is homotopy equivalent to a bouquet of |C(g)| − (e − 1) n+1 spheres S n−1 . 6The aforementioned results represent a strengthening of those in =-=[D2]-=- (in which the homological version of Theorems 1. and 3. above was proven). The next results, contained in sections 5 and 6, use the general approach by minimality from [PS], and significantly extend ... |

1 | Affine complete intersections and Newton polyhedra, Forschungsschwerpunkt Komplexe Mannigfaltigkeiten, Schriftenreihe, heft 70 - Hamm - 1990 |